The oxygen tank consists of a right cylinder with two hemispheres attached to the top and bottom. We need to express the total volume $V(x, y)$ of the tank as a function of the radius $x$ and the height $y$ of the cylinder.

Volume of the Cylinder

The volume of the right cylinder with radius $x$ and height $y$ is given by the formula for the volume of a cylinder: $V_{\text{cylinder}} = \pi x^2 y$

Volume of the Hemispheres

The tank has two hemispheres, one on the top and one on the bottom, each with radius $x$. The volume of one hemisphere is half the volume of a sphere, and the volume of a sphere with radius $x$ is: $V_{\text{sphere}} = \frac{4}{3} \pi x^3$ So the volume of one hemisphere is: $V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \pi x^3 = \frac{2}{3} \pi x^3$ Since there are two hemispheres, their total volume is: $V_{\text{hemispheres}} = 2 \times \frac{2}{3} \pi x^3 = \frac{4}{3} \pi x^3$

Total Volume of the Tank

The total volume $V(x, y)$ of the tank is the sum of the volume of the cylinder and the volume of the two hemispheres: $V(x, y) = V_{\text{cylinder}} + V_{\text{hemispheres}} = \pi x^2 y + \frac{4}{3} \pi x^3$ This is the formula for the volume of the tank.

Finding $V(15, 3)$

Now, let's compute the volume when $x = 15$ and $y = 3$: $V(15, 3) = \pi (15)^2 (3) + \frac{4}{3} \pi (15)^3$ First, calculate each part: $V(15, 3) = \pi (225)(3) + \frac{4}{3} \pi (3375)$ $V(15, 3) = 675\pi + \frac{4}{3} \times 3375\pi$ $V(15, 3) = 675\pi + 4500\pi$ $V(15, 3) = 5175\pi$

Thus, the volume is approximately: $V(15, 3) \approx 5175 \times 3.1416 = 16257.82 \, \text{units}^3$

Explanation

This means that a tank with a cylindrical height of $3$ units and a radius of $15$ units has a total volume of approximately $16257.82 \, \text{units}^3$.

Would you like more details on any part? Here are some related questions:

1. How would the volume change if the radius doubled but the height stayed the same?
2. How can we find the surface area of the oxygen tank?
3. What happens to the volume if both the radius and height are halved?
4. What is the ratio of the volume of the hemispheres to the volume of the cylinder?
5. How does the formula change if the tank is a full sphere instead of two hemispheres?

Tip: For composite shapes like this, break the problem into simpler parts (cylinder and hemisphere) and then sum their volumes.